Since the left hand side is monotonic in \(x\)  and the right hand side is continuous, convergence in (REF ) is locally uniform in \(x\)  (that is, for any \(0<a, b<\infty \)  convergence in (REF ) is uniform on \([a, b]\) ).
See theorem 1.1.6 and corollary 1.2.4 in [1]}.
